Does the Intermediate Value Theorem Apply If h(a) and h(b) Have Opposite Signs?

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Discussion Overview

The discussion revolves around the application of the Intermediate Value Theorem (IVT) to a continuous function h(x) under the condition that h(b) is less than or equal to zero and zero is less than or equal to h(a). Participants explore whether the conditions for applying the IVT are satisfied in this scenario.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions whether the hypotheses of the IVT are satisfied given the conditions h(b) <= 0 and 0 <= h(a).
  • Another participant reiterates the conditions of the IVT, emphasizing the requirement of continuity and the existence of a number u between h(b) and h(a).
  • A later reply suggests that under the assumption that a < b, one can conclude that there exists a c in the interval (a, b) such that h(c) = 0.

Areas of Agreement / Disagreement

Participants express uncertainty about the applicability of the IVT in this specific case, and there is no clear consensus on whether the conditions are fully met.

TheDougheyMan
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I have a continuous function h(x) and the inequality h(b)<=0<=h(a). Can I apply IVT?
 
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Well, what are the hypotheses of the IVT? Are they satisfied in your case?
 
Ackbach said:
Well, what are the hypotheses of the IVT? Are they satisfied in your case?
Isn't it just that h(x) is continuous and if u is a number between h(b) and h(a),
h(b) < u < h(a), then etc etc. I didn't think I could, but I just wanted to see a variation of it.
 
TheDougheyMan said:
Isn't it just that h(x) is continuous and if u is a number between h(b) and h(a),
h(b) < u < h(a), then etc etc. I didn't think I could, but I just wanted to see a variation of it.

Exactly right. So you can conclude that there is a $c \in (a,b)$ (I'm assuming $a<b$) such that $h(c)=0$.
 

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